Download What is topology?

Euclidean Geometry
http://www.youtube.com/watch?v=_KUGLOiZyK8
Pythagorean Theorem
Suppose a right triangle ∆ABC has a right angle
at C, hypotenuse c, and sides a and b. Then
Proof of Pythagorean Theorem
What assumptions are made?
Other proof:
http://www.youtube.com/watch?v=CAkMUdeB06o
Pythagorean Rap Video
Euclid’s
Elements
• Dates back to 300 BC
• Euclid’s Elements as
translated by Billingsley
appeared in 1570
• Ranks second only to the
Bible as the most
published book in history
Euclid’s First 4 Postulates
1. We can draw a unique line segment between
any two points.
2. Any line segment can be continued
indefinitely.
3. A circle of any radius and any center can be
drawn.
4. Any two right angles are congruent.
Euclid’s 5th Postulate
(Parallel Postulate)
5. Given a line l and a point P not on l, there
exists a unique line l’ through P which does not
intersect l.
Distance
Let d(P,Q) be a function which assigns a positive
real number to any pair of points in the plane.
Then d is a distance function (or metric) if it
satisfies the following three properties for any
three points in the plane:
1. d(P,Q) = d(Q,P)
2. d(P,Q) ≥ 0 with equality if and only if
P=Q
3. d(P,R) ≤ d(P,Q) + d(Q,R)
(triangle inequality)
(often write |PQ| for distance)
Euclidean Distance
Let P = (a,b) and Q = (c,d). Then the Euclidean
distance between P and Q, is
|PQ|=√
2
(a-c)
+
2
(b-d)
Taxi-cab Metric
A different distance, called the taxi-cab metric,
is given by
|PQ| = |a-c| + |b-d|
Circles
The circle CP(r) centered at P with radius r is the
set
CP(r)={Q : |PQ| = r}
Isometries and Congruence
An isometry is a map that preserves distances.
Thus f is an isometry if and only if
|f(P)f(Q)| = |PQ|
Two sets of points (which define a triangle,
angle, or some other figure) are congruent if
there exists an isometry which maps one set to
the other
More Axioms to Guarantee
Existence of Isometries
6. Given any points P and Q, there exists an
isometry f so that f(P) = Q (translations)
7. Given a point P and any two points Q and R
which are equidistant from P, there exists an
isometry f such that f(P) = P and f(Q) = R
(rotations)
8. Given any line l, there exists an isometry f
such that f(P)=P if P is on l and f(P) ≠ P if P is not
on l (reflections)
Congruent Triangles: SSS
Theorem: If the corresponding
sides of two triangles ∆ABC and
∆A’B’C’ have equal lengths, then
the two triangles are congruent.
Categories of Isometries
An isometry is a direct (proper) isometry if it
preserves the orientation of every triangle.
Otherwise the isometry is indirect (improper).
Important: It suffices to check what the isometry
does for just one triangle.
If an isometry f is such that there is a point P
with f(P) = P, then P is called a fixed point of the
isometry.
Transformations
1. An isometry f is a translation if it is direct and
is either the identity or has no fixed points.
2. An isometry f is a rotation if it is a direct
isometry and is either the identity or there exists
exactly one fixed point P (the center of rotation).
3. An isometry f is a reflection through the line l
if f(P) = P for every point P on l and f(P) ≠ P for
every point P not on l.
Pictures of Transformations
Sample Geometry Proof
Prove that if the isometry f
is a reflection, then f is not a
direct isometry.
What happens if…
• You do a reflection followed by another
reflection?
• You do a reflection followed by the same
reflection?
Parallel Lines
Euclid stated his fifth
postulate in this form:
Suppose a line meets two
other lines so that the sum
of the angles on one side is
less that two right angles.
Then the other two lines
meet at a point on that side.
Angles and Parallel Lines
Which angles are equal?
Sum of Angles in Triangle
The interior angles in a triangle add up to 180°
What about quadrilaterals?
More generalizing
• What about polygons with n sides?
• What about regular polygons (where all sides
have the same lengths and all angles are
equal)?
Exterior Angles of Polygons
Another Geometry Proof
Theorem (Pons Asinorum):The base angles of an
isosceles triangle are equal.
Symmetries of the Square
A symmetry of a figure is an isometry of the
plane that leaves the figure fixed. What are the
symmetries of the square?
The Group of
Symmetries of the Square
The set {a,b,c,d,e,f,g,h} together with the operation
of composition (combining elements) forms a
group. This is a very important mathematical
structure that possesses the following:
1. Closed under the operation
2. The operation is associative (brackets don’t
matter)
3. There is an identity element
4. Every element has an inverse
Frieze Groups
• A frieze group is the
symmetry group of a
repeated pattern on a
strip which is invariant
under a translation
along the strip
• Here are four
possibilities. Are there
any more?
Frieze Groups
Wallpaper Groups
• Symmetry groups in the plane
• Show up in decorative art from cultures
around the world
• Involve rotations, translations, reflections and
glide reflections
• How many are there?
Similar Triangles
AB/DE = AC/DF = BC/EF
Pentagon Exercise
Which triangles are congruent?
Isosceles? Similar?
The Golden Ratio
The golden ratio is defined to be the number Φ
defined by
Φ = (1 + √5)/2 ≈
1.618
Golden Pentagon
What is the ratio of your
height to the length
from the floor to your
belly button?
The Golden Rectangle
The Golden Ratio
A
B
Ratio of A to B is the golden ratio
A = 1.618 B
Where can we find the Golden Ratio?
The Parthenon
The Great Pyramid of Giza
2560 BC
-Side lengths approximately 230m
-Base covers 53 000 m^2
-Sides angled at 51.5 degrees.
1^2 + (√φ)^2 = φ^2
√φ
φ
1
2
1+ 1.618 = 2.618
CN Tower
Base to observation deck 342 m
Base to spire 553.33 m
553.33/342 = 1.618 = φ
Moving on…
1,1,2,3,5,8,13,…
What is the pattern?
Fibonacci Numbers
• Each number is the sum of the two
before
Fn= Fn-1+Fn-2
• Fibonacci Numbers in Nature
• Youtube video on Fibonacci
Ratios of Fibonacci Numbers
•
•
•
•
•
F2/F1=1/1=1
F3/F2=2/1=2
F4/F3=3/2=1.5
F5/F4=5/3=1.67
F6/F5=8/5=1.6
•
•
•
•
•
F7/F6=13/8=1.625
F8/F7=21/13=1.6154
F9/F8=34/21=1.619
F10/F9=55/34=1.6176
F11/F10=89/55=1.6182
Connection
• So the Fibonacci numbers and
the golden ratio are connected
• More about the Fibonacci
Sequence and The Golden Ratio
Pascal’s Triangle
More on Pascal’s Triangle
• All You Ever Wanted to Know
About Pascal's Triangle and more
• Pascal's Triangle and the
Fibonacci Series
Connection
• So the Fibonacci numbers and
Pascal’s triangle are also
connected!